Appendix E
E.1 Stochastic Dynamics
The stochastic differential equations, known also as Langevin Equations can be
with aditive or multiplicative noise. The stochastic infinitecimal used in their interpretation and solution has different rules than those used in the usual infinitesimal
calculus. The Stochastic Dynamics term refers to the temporary evolution of random
variables subject to noise. We give here the basic konwledge on this subject. For
more knowledge and examples consult the bibliography.
E.1.1 White Noise and Wiener Process
A stochastic process (SP) very important for the “Stochastic Calculus” is called
“White Noise”, ξ(t), defined by its statistics properties:
ξ(t) = 0
ξ(t 2 )ξ(t 1 ) = δ (t 2 − t 1 )
E.1.2 Spectral Intensity
The electromagnetic signals emited by electronic circuits have stochastic components, called noise. we will consider stationary signals, of mean null, X = 0. The
intensity of the frequency component ω of the signal X(t), called spectral intensity,
I (ω), is related with the auto-correlation function, X(t + τ )X(t), through the
Wiener-Khintchine theorem, namely
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in
Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9
165
E.1 Stochastic Dynamics
The stochastic differential equations, known also as Langevin Equations can be
with aditive or multiplicative noise. The stochastic infinitecimal used in their interpretation and solution has different rules than those used in the usual infinitesimal
calculus. The Stochastic Dynamics term refers to the temporary evolution of random
variables subject to noise. We give here the basic konwledge on this subject. For
more knowledge and examples consult the bibliography.
E.1.1 White Noise and Wiener Process
A stochastic process (SP) very important for the “Stochastic Calculus” is called
“White Noise”, ξ(t), defined by its statistics properties:
ξ(t) = 0
ξ(t 2 )ξ(t 1 ) = δ (t 2 − t 1 )
E.1.2 Spectral Intensity
The electromagnetic signals emited by electronic circuits have stochastic components, called noise. we will consider stationary signals, of mean null, X = 0. The
intensity of the frequency component ω of the signal X(t), called spectral intensity,
I (ω), is related with the auto-correlation function, X(t + τ )X(t), through the
Wiener-Khintchine theorem, namely
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in
Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9
165